Reading user642796 (https://math.stackexchange.com/users/8348/user642796) at https://math.stackexchange.com/q/243133 I believed I finally figured out what might be a model of a theory, to evaluate my understanding I set this example, could you please confirm or infirm my words?
- $\mathcal{T}$ is the theory of rings, a set of theorems (which are sentences : well-formed formulas with bounded variables) all interpreted as true. From the elements of a subset of theorems called axioms we can logically infer other theorems called …theorems.
- $\sigma_\mathcal{R}$-structure over $\mathrm{R}$ (the set of reals) is a model of $\mathcal{T}$ if and only if :
- is signature $\sigma_\mathcal{R}$ is the same as the one of $\mathcal{T}$
- each sentence in $\mathcal{T}$ is satisfied by those of the $\sigma_\mathcal{R}$-structure over $\mathrm{R}$ (in mathematical jargon it is denoted $\mathcal{M} \models \mathcal{T}$) which means that if we interpret the theory $\mathcal{T}$ into the model $\sigma_\mathcal{R}$-structure over $\mathrm{R}$, the truth values (which are all true) of sentences in $\mathcal{T}$ are preserved in $\sigma_\mathcal{R}$-structure over $\mathrm{R}$ (I don't figure out yet if it is a one-to-one or one-to-many map)
The theory of rings $\mathcal{T}$ is composed of :
The domain of discourse $\mathrm{D}$ (the set of entities over which the variables may range : this is all the elements which doesn't contradict the theorems)
$\sigma_\mathcal{T} = \{0^\mathrm{D},1^\mathrm{D}, +^\mathrm{D},\times^\mathrm{D}\}$ is the signature of the theory also called language (not to be conflate with the formal language of the theory which is all the well formed formulas of the theory) where :
$0,1$ are nullary symbols of function also called symbols of constant
$+,\times$ are the binary symbols of function
$0^\mathrm{D},1^\mathrm{D}, +^\mathrm{D},\times^\mathrm{D}$ are the interpretations of those non-logical symbols in $\mathrm{D}$
The set of axioms given for $0^\mathrm{D} = 0 \in \mathrm{D}$, $1^\mathrm{D} = 1 \in \mathrm{D}$, and $\forall a \in \mathrm{D}, \forall b \in \mathrm{D}, \forall c \in \mathrm{D} $ (taken from wikipedia) :
$(a+b)+c = a + (b+c)$
$a + b = b + a $
$a + 0 = a$
$a + (-a) = 0$
$(a \times b) \times c = a \times (b \times c)$
$a \times 1 = a = 1 \times a$
$a \times (b+c) = (a \times b) + (a \times c)$
$(b + c) \times a = (b \times a) + (c \times a)$
Using those axioms we can infer other theorems like :
- $a \times 0 = 0$ because $a \times 0 = a \times ( a + (-a)) = (a \times a) + (a \times (-a)) $ etc
The $\sigma_\mathcal{R}$-structure over $\mathrm{R}$ has :
- $\sigma_\mathcal{R} = \sigma_\mathcal{T} $ is the same as the one of $\mathcal{T}$
- each sentence in $\sigma_\mathcal{R} = \sigma_\mathcal{T} $ satisfied those of $\mathcal{T}$ when we substitutes the symbols $a,b,c$ by those of $\mathrm{R}$ the theorems of $\mathcal{T}$ hold, therefore if we restrain $\mathrm{D}$ to $\mathrm{R}$, the theorems hold.
That is what I think of a theory, a model of this theory and their relation, what do you think of that?
I also think we could set an alternative point of view by using the domain of discourse in interpretation of the theory in a model, therefore a theory to be interesting should have necessarily a model which satisfied the sentences of the theory : the element $a,b,c$ of the domain of discourse will no longer be viewed as elements of the domain of discourse, but as non logical symbols which belongs to the theory, there, when we quantify them, the quantification acts as a one to many function : $a$ range over $\mathrm{D}$. This point of view allows us to set the theory independently of any domain of discourse, but to verify if the theory is interesting we will have to interpret it in a model, I don't know if i'm clear here, anyway, it's not the core of my question.